calculus

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i understand optimization but im stuck on this problem.. mrs.day is building a new deck. she has 580 square feet to enclose. if one side is bounded by a wall, find the minimum cost to build the deck if one pair of opposite sides cost $12 per foot and remaining sides cost $19 per foot. what are the demisions of the deck?

  • calculus -

    Is she building a deck or a railing around the deck.
    From the wording I will assume a railing.
    The two sides of the railing, length a, cost 12/ft each.
    The front of the railing, length b, costs 19.ft

    a*b = 580

    2*12 a + 19 b = c, the cost

    dc/da = 0 for min = 24 da/da +19 db/da
    but
    b = 580/a
    db/da = -580/a^2
    so
    0 = 24 +19 (-580)/a^2
    24 a^2 = 19*590
    a^2 = 459
    a = 21.4
    then b = 27
    Now it might be that the front cost 12 and the sides cost 19
    in that case
    2*19 a + 12 b = c
    dc/da = 0 = 38 da/da + 12 db/da
    db/da is still -580/a^2
    38 = 12(580)/a^2
    a^2 = 183
    a = 13.5
    then b = 42.8

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